PFC/JA-92-16
Bootstrap Current Induced by Fusion Born
Alpha Particles
C. T. Hsu, K. T. Shaing,* D. J. Sigmar, R. Gormley
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139 USA
This work was supported by the US Department of Energy under contract DE-FG02-91ER54109. Reproduction, translation, publication, use, and disposal, in whole or in part, by
or for the US Government is permitted.
Submitted for publication in: Physics of Fluids B
* Oak Ridge National Laboratory, Oak Ridge, TN 37831
BOOTSTRAP CURRENT INDUCED BY FUSION BORN
ALPHA PARTICLES
C. T. HSU, R. P. GORMLEY, D. J. SIGMAR
Massachusetts Institute of Technology, Plasma Fusion Center, 167 Albany Street,
NW16-260, Cambridge, MA 02139 USA, Tel: (617) 253-2470
K. C. SHAING
Oak Ridge National Laboratory, Oak Ridge, TN 37831
Abstract
The bootstrap current produced by fusion born alpha particles is obtained retaining
effects of slowing down drag, pitch angle scattering, and arbitrary aspect ratio.
The
result is presented both as a summation of a rapidly converging series and a simple Pade
approximation good for arbitrary aspect ratio.
Quantitative results are derived using
International Thermonuclear Experimental Reactor (ITER)' parameters.
1
I. INTRODUCTION
In toroidal plasma systems, the parallel friction force will attempt to equilibrate the
parallel flows of different species by generating a relative poloidal flow in response to the
relative parallel diamagnetic flow. A finite parallel relative flow remains due to existence
of magnetically trapped orbits since they cannot contribute to the poloidal flows. This
effect leads to a current, usually referred to as the bootstrap current which is proportional
to the relative parallel diamagnetic return flow times the fraction of trapped particles.
The neoclassical bootstrap current is of great importance for future tokamak fusion
devices because at sufficient poloidal beta it can provide a substantial fraction of the
total current. Typically, the fusion produced fast a particles can produce a beta gradient
comparable to that of the background plasma. This implies that fast a particles may result
in a bootstrap current contribution comparable with the background bootstrap current.
Previously, the a particle induced bootstrap current had been calculated 2 3 by keeping
only the drag term in the collision operator and was found to be small because it arose
only due to "banana collapsing" during the slowing down process, which scales as oc 3/2
(f is the inverse aspect ratio). However, in recent work, 4 it has been found that pitch angle
scattering can be very important in the neoclassical transport process of fast a particles
and leads to transport fluxes oc e1/2. In this work, the bootstrap current will be calculated
based on the work of Ref. 4 in which both drag and pitch angle scattering collisional effects
are included for the a particle kinetics.
By imposing the quasineutrality condition, the plasma current can be written as
J
=-,
e[neVie + %lZctVcf+
Z njZ Vjj
1
(1)
where the subscript i refers to the main ion, and I refers to impurities and Vij = Vi - Vj
is the difference of flows between species i and j. Hence, the current can be obtained upon
determining the relative flow velocities Vi, Vai, and Vji.
It will be shown that the dynamics of all bulk plasma ion species (i.e., i, I) are hardly
influenced by a particles, and thus can be treated as known quantities which can be found
2
Thus in the next two sections, only the a particle parallel
in the existing literature.
flow and electron parallel flow will be calculated. In Section IV, the bootstrap current
will be calculated and the quantitative results will be given based on ITER parameters.
Conclusions are given in Section V.
II. CALCULATION OF Vi
In Ref. 4, the a particle response was calculated by including the effects of finite aspect
ratio and both the drag and pitch angle scattering. Nonetheless, ion flow and E x B drift
was neglected. Although this can be justified due to the fact that
Vi
Va
T
Ea
it is still interesting to include them (i) for completeness and (ii) to demonstrate that the
a transport flux is independent of E,.
By considering also the fact that the fusion source term is isotropic in the ion rest
frame instead of the lab frame, one finds that it is appropriate to treat the a particle kinetic
equation in the ion rest frame. In Appendix A, the a drift kinetic equation in the ion rest
frame is obtained. The resulting drift kinetic equation (A8) is then solved by following the
same procedure as in Ref. 4 (also cf. Appendix A) which yields the first order distribution
function
fa
foo + v 11
-
fao + P(Aw,')
(2)
with
An (A, ?) V,,i(w,,O)
P(A, w,O)=E
j=1,3
A (W,"o).
(3)
n=1
B ;
h;h
2
2;
Here, v is the a particle velocity in the ion rest frame, w
2
ZjeB I
B .R VO. V*
L9~
+ KiB is the ion parallel flow less the E x B
=
;*i
B mc
drift induced return flow; Ki
corresponds to the ion poloidal flow; pi is the ion
A
pressure; An is the eigenfunction of the pitch angle scattering operator as given in Ref. 4;
and
fo(V,)
=
S. 3
47r(v3 + V3)
3
H(vo - v)
(4)
is the lowest order (slowing down) distribution function driven by the fusion source S.
P(A,w, ,) represents the localized distribution function which vanishes in the trapped
region.
Denoting the flux surface average by
Al
( ), the
- IV
three driving forces for P(A, w, i)
9 faO
(5)
1 8
A2
(6)
57- foaO
A3
ea
are
2 9 fao,
)
(7)
and the corresponding V,, are (cf. Appendix A)
Vni = On
Vn2= On
-
(K, - 1)V3
b
(n 1)V3
1 -
VO
du
Here,
Qp, Kn
-4n
(v3+vi)9
c
3(v3
VO
~ 3s
and on are given in Ref. 4; in particular,
E
is the effective parallel electric field; F,
=
OfCO(U)
(8)
Qpr-)
3'C
U)(9)
V 3)v Qu+) 3
du v 3 (u3 + )
U3 (v 3 + V3)
Jv
)
3
bV(+U du
(n-
++
VO
=
(v3(V3 +U3)
E=l -
nie
U2 afo(u).
(10)
OU
Q=, v
/V3
-
0(1) for D-T fusion.
ZF
(11)
is the electron-ion parallel friction and
In
nSr
+-
In v)
f'o.
(12)
V
It is now apparent that fai is independent of the E x B drift. In addition, all the terms
involving ion flow are
-
smaller than the a particle diamagnetic drift terms. This justifies
the adequacy of the results in Ref. 4 and rigorously proves that the neoclassical a particle
radial fluxes do not explicitly depend on E,.
4
Now, Vj can be calculated from
ncV,
dvvllf
1 =J
0,
1 =
vadj
dA
- no
nh ))
fci
(13)
which yields
n, 1
1-
=
n= 1 KnJ
1
1 )
-Kn
[-Yn (1 -
T2
(U
n=1
U2n - N2n
(14)
E\
+ N 3 Za erS
h
Ma
Here, a = vil /1vl 1 , -yn is defined in Eq. (17) of Ref. 4,
U1
a
1
-
fc1]
[
dV 3
Tdv- ?j-fo=
P
L
=2
SrvoE~LjXt
(15)
is the a diamagnetic flow,
U2 n
H,(v)
4
N2n =
N 3 =E
Gn(v)
dv [Hn(v) +
1 jU
X2
f-5
Gn(v)) fao,
3
u3 (V 3 + va3
1 du
3(U3 + V3)
dx (
U3
U3 + v3,'
x3 -
(16b)
(17)
Hn(v)] fo0,
nJ dv Gn(v) + v
Lga33(1-1) f
X,
u3(ui~})Qp-n
du
00
(16a)
fo)
H(v) (
JdV
(18a)
(18b)
(19)
(In Sr,),
(20a)
(ln v),
(20b)
5
and X0 = !
is the ratio of a birth to critical velocity.
Here, it is important to note some properties of -y,
and
it,
fe,
the effective fraction
of circulating particles, 5 ie,
A dA-
S-
=(
(21)
2
which corresponds to the "pitch angle scattering" dominant solution of the a drift kinetic
equation.
7n =-d
(h
n=1
corresponds to the "drag" dominant case.
a
=
2
)
(22)
is a pitch angle variable.
In addition, Hn(v) and Gn(v) have the following properties:
Hn(v), Gn(v)
for
= 1
for Qpia = 0
for Q-n > 1.
~
Also, using the fact that Kn > 2n 2
-
QPs > 0
< 1
0
(Q
)
n and typically Q
-
(23)
1, one expects a rapid con-
vergence of Hn(v) and Gn(v) with n. This implies that one needs only the first few
eigenfunctions (e.g. n = 1 -+ 5) to accurately evaluate the second and third terms in
Eq. (14).
Furthermore, when (i)
Q >
1, (i.e. pitch angle scattering dominant), only the first
Q, <
term in Eq. (14) survives; and when (ii)
K
~>(
njVail')
(-
+
(
1
-n
1, Eq. (14) reduces to
'>n) (Ui-nc
**n=1~
n Zer
n=1
as is expected (also cf. Eq. (22)).
Thus, the a dynamics has been solved by including the effects of ion flow and the
effective parallel electric field. However, these additional effects are insignificant quantita-
tively because (i) they are 0
(
) smaller than the
6
a diamagnetic terms, and (ii) they
contribute only 0
(v"
corrections to the conventional bootstrap current (as will be
shown in the next section). One can thus simplify Eq. (14) by omitting these effects, and
we obtain
nc,
= ftp U1 -'
(Yn (1 -
U2n
,(24)
n=1
where
f
=
1 -
flP
corresponds to the fraction of trapped particles where p denotes the
pitch angle dominated solution, see Eq. (21).
Furthermore, although the second term of Eq. (24) can be evaluated using a few
numerical eigenfunctions, it is highly desirable to write it in a simpler analytic form.
Using the asymptotic behavior of U2n at Q -- 0, Q >> 1, xo -- 0, Xo >> 1, a Pad6 form
is obtained in Eq. (B6) of Appendix B. Equation (24) can thus be written as
2:
h
=
Srvo
fd
+ Q L
L
Qp=t + 4fc'L-j
t
- f/d)
LtXt.
(25)
The first term in the bracket is clearly due to the drag and the second term is due to the
pitch angle scattering effects. Note that for c < 1, ftd ~ 1.6E3/2, and fP ~ 1.46e1/2. Since
f/
~lf , it does not require a very large
Q,
to make the pitch angle scattering effects
important.
Furthermore, the Padi approximations of fd,
f4
for arbitrary E are obtained in Eq. (B9).
Note that, as shown in Fig. (1), Eq. (B9) is in excellent agreement with the exact numerical calculations. It is also worth mentioning that by using Eq. (B9), the simple analytic
Pads form Eq. (25) has been compared with numerical results evaluated from Eq. (24).
Remarkably, over wide ranges of E, xo, and
Q,,
the difference is always within 10%. This
actually implies that the error of Eq. (25) is much less than 10% (cf. Appendix B).
7
III. CALCULATION OF Vej
First, using the facts that Ime > 1 and
V1-
Vthe
< 1, one can obtain an approximate
e - a collision operator in the ion rest frame as
3eaVtflaZ3he
4
7 neZeff
Cea = -
8
&v
U -U
f, +
2V2
Vfe
-v.i
3fem
the V
.
That is,
vlh
3aZ
3
t
Cea + Cei= -o-- re
4
a f,
+
-- U --8v
Ovf
+ ne
2 and the equality
vMv,,rZeff = Ej ni n.
Here, U= ----
nZ, 2Vi-
+cinZZe
m
Yef
2V3he
fe]
theleef V
(26)
y
= T has been used.
Note that the first term in Eq. (26) corresponds to pitch angle scattering and that
the second term is a momentum input to the electrons due to Vi.
Also note that the
a contribution to pitch angle scattering is insignificant because it simply replaces
y
+
+ne, Z2
Vai is 0
) and
_z__)
Furthermore, since
n.Z.
<
with
1. On the other hand, the momentum input term involving
compared with the first term in Eq. (26), and thus can be significant.
, which can be at most of order one in a typical
F
fusion device, one can conclude that ion and impurity dynamics will hardly be modified
by alphas because (B - V - Hi) > (BFeiii)
(BFec,1)
(BFaii).
To solve the electron dynamics the moment approach as developed in Ref. 5 will be
adopted in order to retain generality with respect to the electron collisionality. In addition,
because the parallel flow viscosity and heat viscosity solely depend on the poloidal particle
flow, heat flow and collisionality of the individual species, both viscous forces do not change
their form in the presence of a particles, except for a trivial increase-of vi
vei =
v
+ n,
.
(or Zff), i.e.,
Therefore, the viscous forces obtained in Ref. 5 can be used here
eff/
without modification.
The moment equations to be solved are the electron parallel momentum equation
(B - V - He) = -nee(E 1 B) + (Fe1 B)
8
(27)
and the electron parallel heat momentum equation
(28)
(B - V - 0,) =_(F(' B),
where
++ = dv
F(2)
dV mV (v
)(vv
_ 5V2
(v2
2
-
2
3
I )fe,
V2he)Ce(fe).
-
2
V
-
Using the two-moment approximation 5 ,6 and Eq. (26), the frictional moments can be
written as
Fe = menevei
F
q,
(VI + 35 ~neTe
neTeveri
-
where V, = Vie + nfle~anZ2 Vi.
8
(V +
(29)
(v'ee +
V
e
)
(30)
The viscous forces 5 are
2'\
+-+4
(B - V - I[e) = men, (B 2) (Li V, + 5 p2qpI
(B -V - 9e) = neTe(B 2 )
(A2 IP +
2
(31)
\
5 /13 p
.
(32)
Here,
Ve = VB +wi R 2 V5
(33)
qe = neTe (qpB + W R2VO)
2
(34)
B
5
W2
B
- 2m
[ pe - ne e
8
4Te,
<,
(35a)
(35b)
and fitted Pad4 forms of 14,42, 13 for all collisionality regimes can be found in Ref. 5.
Note that in the banana regime
isi - f(vi
+ 0.754vee),
9
(36a)
SP
-- 7- (1. 51vi + 0. 884v,,),
pA2 ~Nf'I
A3
~
fcp
(36b)
(36c)
(3.25vi + 1.94ve,).
Concerning Vai, it is now clear that the terms involving (T)
are 0 (nZ
)
compared with (T)
and ()
in Eq. (14)
terms in Eqs. (29) and (30), and (E1 IB) terms in
Eq. (27) respectively. Equations (27)-(36) thus yield
(VicIIB) = (V0
B)
1+ 0
n( Z)
tellne
- (1 - FZ) n
Zeff
f ViII B),
(37)
ne Zeff
where
[=Z3 + 2(vee + 1vei)] - A2(A2 - 2vei)
(Al + Vei) [/3 + 2(vee +
Vei)] - (A2 - Svei)2
A"
is the effective fraction of trapped electrons; and (V ilB)
= Bo([) can be evaluated
using Eqs. (24) and (25). Note that the superscript (0) in Eq. (37) refers to the conventional
results without the contribution from a particles. Also, from the fact that ftp <
large aspect ratio (E
<
1), one has pj
<
vei, and F. -+ 0. Consequently, for E
"
(ViIIB) =(V.(OB) 1 + 0
_(aZ
elnze
neZeff
<
f'
at
1,
(Vai IIB).
This can also be understood from the fact that for E < 1, the poloidal flow V, of the
electrons is solely driven by the ion poloidal flow via electron-ion friction; and the poloidal
heat flow qp is solely determined by the electron diamagnetic heat flow. This implies that
the total electron friction is unchanged by the alphas. Therefore, (V, 1 1B) will be reduced
simply according to the momentum input from
U
B) via electron-a friction (see also
Eq. (29)). This also implies that, for E < 1, the electron radial flux is not changed by a
particles. However, for generality, Eq. (37) should be used to include the finite e effects.
10
IV. BOOTSTRAP CURRENT
It is now straightforward to calculate the total parallel current from Eqs. (1) and (37).
This yields
(J11B) = (J(9)B)
11
1+ 0
(lZ
n, Zeff
) + nZee (1 -
Z
Zef
(1 - F)) (V1
B),
cl1 ),
(39)
(9
where (V,, 11B) is given in Eqs. (24) and (25). Eqs. (24), (25) and (39) thus yield the
bootstrap current produced by alphas as
J,'=
Here, the term involving
1-
Z
(1 -F)
P.
c
Fa
(40)
in Eq. (40) corresponds to the well known electron screening
effects,5, 7 and
fU. - (h
2
) Z
yn(1 U1
L)U 2 n
~
fF2+ EZ=1
,
L4
LdXt
_1 (ff-))
1=1 LtXt
(41)
is the effective fraction of trapped alphas.
Note that the second term in Eq. (41) involving
scattering. When
Q,
Q,
-+ 0 or vo/vc > 1 (in which case Lt/Lt+ > 1), this term becomes
negligible and the result reduces to the drag-only result
~ 1, and
ft/ftp
-
is due to the effects of pitch angle
ft.
However, since
Qp
is always
e, it is not difficult to make the second term in Eq. (41) significant. In
Fig. (2), results for F," are shown for different vo. The asymptotic case E0 = oo refers
to the drag-only result. It is seen that F increases with decreasing vo. The reason all
expressions reduce to the drag-only result at r/a = 1 is that
because the temperature profile is taken to be To(1 - r 2 /a
2
yc
= 0 at this point. This is
)V,.
In order to evaluate the bootstrap current quantitatively, it is essential to self-consistently
solve for the equilibrium after incorporating Jbs,(/) into the parallel Ampere's law. In this
work, for simplicity, we assume circular geometry, i.e,
I a
,
11
1 a
The parallel Ampere's law thus becomes
18
r or
4tr
rBp = -
c
(
JOH + J()
+
(42)
)
Here,
(43)
JOH = F,&O'Spitzer(EIl /h)
is the ohmic current;
j(O)
ba
-\B0
-c )al F
Pe + Pi) - FeFi
aTe
a Ti - FTne 19rj
(44)
is the background bootstrap current;
(13 + 2(ve + 1-vei))vei
(iL1
+ Vei)[A3 + 2(vee + 8 vei)] - (P2 -
vu)2
describes the neoclassical correction to Spitzer conductivity,8 and
FT -
vL'ei) - 1.5I3vei
-2/12(vee +
(1 + vei)[IL3 + 2(vee + L3-vei)] - (P2 -
46)
Ivei)
FZT is obtained by letting vi -- 0 and replacing vee -+ vi in Eqs. (46) and (36). For ions
in the banana regime with e
<
1, F~ y~ 1.17. Note that terms involving F
are due to
the thermal friction force.
Equations (40)-(46) are solved numerically by adopting ITER parametersi at the
steady state phase: (ne) = 0.64 x 10 2 0 m- 3 , (T,) = 20 KeV, Ip =19 MA, K = 2,,3p = 1.1,
a = 2.15m, Ro = 6m.
Te oc (1 - r 2 /a
2
Also, the radial profile is assumed to be n. oc (1 - r2/a2),,
)"T with vn = 0.5,vT = 0.75, and the inductive parallel electric field
(Ell /h) is chosen so that the total plasma current including the bootstrap currents reaches
Ip ~ 19MA. For F,,
ff,
and f', the Pad4 forms given in Eqs. (41) and (B9) are used.
Also, both electron and ion species are assumed to be in the banana regime throughout
the plasma. The effect of elongation is considered by replacing the plasma minor radius
by av 4r.
First, with Zeff = 1.5, the resulting poloidal magnetic field and bootstrap current
densities are shown in Figs. (3)-(5).
One notices the slightly decreased Bp due to
12
J,.
The alpha induced bootstrap current amounts to ~ 0.385MA, which is ~ 7.2% of the
background bootstrap current.
Varying Zeff, one finds a strong dependence of Jb, on Zeff. In fact, the effects of
Zeff on J, are threefold:
(i)
The electron screening effect decreases with increasing Zeff. Indeed, for Zeff < ZC,
at e
<
1, F. becomes negligible and J. becomes negative.
(ii) Alpha pitch angle scattering effects increase with Zeff (because Qp oc Zeff). That
is, F, increases with Zeff.
(iii) However, the a pressure decreases with Zeff due to fuel dilution. Actually, when
Zeff (r) is highly peaked near the axis, both Pth(r) and Pa(r)can become flat or even
hollow; and the bootstrap current density Jb, can be significantly reduced.
Table 1 shows the final currents for two cases where the Zeff profile is flat; one
where Zff = 1.5; the other where Zeff = 2.2. The ohmic seed current is kept fixed at
13MA. For Zeff = 1.5, the background bootstrap current:: 30% of the total current. The
calculated alpha induced bootstrap current reaches ~ 7.2% of the conventional bootstrap
current. When Zff is increased to Zeff = 2.2, the total current decreases slightly by
about 2%. On the other hand, the conventional bootstrap current decreases by 5% while
the alpha bootstrap current decreases by 21%. Although a larger Zeff has the effect of
diminishing the temperature screening term in the alpha bootstrap current (see Eq. (40)),
and increasing the alpha pitch angle scattering, increased fuel dilution has a dominant
effect.
Specifically, when Zeff is increased, causing more fuel dilution, the alpha and
thermal pressure decreases resulting in less bootstrap current. Since the dilution effect is
squared in the alpha pressure while occurring only linearly in the thermal pressure, the
effect on the alpha bootstrap current is pronounced.
The effect of having a peaked Zeff profile was also investigated. Here we let Zeff (r)
0.7exp[-16r2 /a 2 ] + 1.5 making Zeff = 2.2 at the the center and 1.5 near the edge. Again,
the dominant Zeff effect is fuel dilution. The resulting currents for this case are listed in
Table 1 and can be seen to be quite similar to the flat Zeff = 1.5 case. This is because,
in the peaked case, Zff rises significantly above 1.5 only near the center where there is
13
little bootstrap current anyway. Note that both the background and the alpha induced
bootstrap current densities become slightly negative near the center. The reason is that
the fuel dilution gradient (specifically d(Zeff)/dr)) has a diminishing effect on the negative
thermal and alpha pressure gradients, and at some small value of r/a causes the bootstrap
current to change direction.
On the other hand, with the fuel dilution effects switched off, the ratio of alpha to
background bootstrap current density is given in Fig. (6), for both the flat Zeff = 1.5 and
the centrally peaked Zeff(r). It is shown that the alpha induced bootstrap current density
increases with Zeff due to the increasing alpha pitch angle scattering and decreasing
electron current screening.
Furthermore, with Zeff = 1.5, by artificially switching off the effects of fuel dilution
and pitch angle scattering individually, as shown in Table 2, the significance of these effects
can be clearly observed. It is found that the effects of pitch angle scattering enhance Iba,
only by 30% while the fuel dilution can reduce Ib by 20%. The reason is that the pitch
angle scattering is most important near the center where the bootstrap current density is
small.
Equation (40) is good for arbitrary vo /vc and Eand is valid for evaluating the bootstrap
current induced by any hot species containing sufficient isotropic pressure. One concludes
that the enhancement of the bootstrap current contribution from energetic ions including
pitch angle scattering will drastically increase with smaller vo.
V. CONCLUSION
The bootstrap current induced by fusion born a particles has been calculated for
general electron collisionality. Our result retains the effects of both a pitch angle scattering
and slowing down drag for arbitrary aspect ratio. The results are presented in Eqs. (40)
and (41), in both forms: a summation over a rapidly converging series E'
I yn(l -
-
)U 2 n
and a simple Pad6 approximation good for arbitrary PQ and c. Convenient Pad6 forms of
the effective fraction of circulating particles
fg
14
and
fe' are
presented in Eq. (B9). It has
also been shown that the radial electric field E, does not play a role in a particle transport
[cf. Eq. (A5)].
The effects of Zeff on Ja are significant because when Zeff increases the electron
screening effect decreases and the a pitch angle scattering effect increases but the a pressure
decreases due to fuel dilution which has been found to be a strong effect. Actually, the
reason that a pitch angle scattering enhances Ib only by 30% is due to the effects of
electron screening and strong fuel dilution.
When Zeff(1 - F") < Za, JU/JE,
can
become negative.
The theoretical results are evaluated quantitatively by using the ITER parameters
and numerically solving the parallel Ampere's law including the bootstrap current terms
oc B'~1 for the self-consistent Bp(r). It is found that the bootstrap current produced
by alpha particles reaches ~ 0.385MA and is about 7.2% of the conventional bootstrap
current.
The results given in Eqs. (40) and (41) are valid for evaluating the bootstrap current
induced by any hot isotropic species. For an anisotropic hot species generated by neutral
beam injection or rf heating, the induced bootstrap current can be calculated by straightforwardly extending the present approach to also include the eigenfunctions even in v11
(work in progress).
15
Acknowledgment
This work was supported by the US Department of Energy under Grant DE-FG0291ER-54109.
16
Appendix A
Derivation of ft ,
Since the source term of fusion born alpha particles is isotropic in the ion rest frame,
we treat the alpha kinetic equation in this frame.
Starting from the kinetic equation,
changing variable v -+ v - Vi and then performing the gyro-average 9 the a drift kinetic
equation to first order in gyroradius becomes
a9
VofVIf, +vd.
(I )
/ b-
9
iVfao+Fo+
+ 2V
Here, fa = fa(A, w,
(
F miZa _
F maZi
b.W .b)
fO = Ca(ya) + 4
-V
_
-_.
2
- vO).
(Al)
Vp;+V - fii-Ri)
(A2)
2(V
,0);V is performed at constant (A, w);
(
(at
+
Vi+V xVV;
±
Z
maZinik
vd = vilb x Vv is the drift velocity in the ion rest frame; and
1'
+ (VVi)T)
Wi((Vvi)
2
_v
1
3 V. V) I(M
3
3)
is the ion velocity strain.
Note that Eq. (Al) agrees with the results by Hazeltine and Ware9 except for the
definition of the F force term. This is because the results in Ref. 9 are for the drift kinetic
equation for a given species in its own rest frame. Furthermore, Eq. (Al) also agrees with
Eq.(5.45) of Hinton and Hazeltine1 0 which is the electron drift kinetic equation in the ion
rest frame.
From Eq. (A2) and Vi = KiB + wiR 2 V. one finds
Fo=
Z
maZini
Vp;FlI,
\=
ZaE,V-Vi=O,b-Wi-b=Kib-VB.
ma
(A4)
Equations (Al) and (A4) thus yield
- + I, 5fao - VI1* -af±fW
O +
+
VI Vil fa
17
Zfev
E
C, 11 Elv
;-fO
(M)
()
S
J 4irv 2 5(v - vo).
C fc)+
=
Let us now turn to the a collision operator in the ion rest frame. By using
a
<
1,
< 1, and Eq. (26), one finds
Cae
-
N
T 8
a ffa ) +
18
T,
Vfa + mc
8v~
ma &V
(A6)
- - fco.
T.nFei
av
r.9mene
Here, Fei is the electron-ion friction. Similarly, Ci can be written as
3
Vb 8v U
2-r, av
CcQt -
8
---fa + V3
,,
8
v
C- . 9V V3 f
(A7)
.
Equations (A5) - (A7) thus lead to
v1 1 |VV
1
8
5
IV ~affaO -v,,V
f'
I*
I
fy1
. (1+
-
=
7ci +
vjcVf- C+
2 8v U
J+
ZCe
18
~vEifcro
- a]
S
+ 4r
2
(A8)
5(v -vo).
Then, by following the same procedure as in Ref. 4, one obtains Eqs. (2) and (3) with the
governing equations for Va3 :
f.O) (0n -
3
3
v 6b/a
3
'
fO)
(0n -
nVai) =r
KnVn2)
f.0 X.V3 + V3 (
=
(~v3 + v)v (ofaO)
a
19V
(3
o)
+
=
which leads to the solutions given in Eqs. (8)-(10).
18
)
19V
[(v3
(A9)
(on - Vn1)]
faO I (On
-
,
(A10)
Vn3]7
(All)
Vn2)
a
8vI7f'O)
I
Appendix B
Pad6 Approximation
The right hand side of Eq. (24) is rigorously expressed by summing over a rapidly
U2 . and can be evaluated quite accurately by using only a
converging series -yn (1 -
few eigenfunctions. However, it is highly desirable to get a simpler analytic form good for
all xo,
Qp,
and e. Let's start with U2 n, which can be written as
2
U2n = S-rvo
(B1)
ZDnI X,
e=1
where
1
D
3X
(B2)
Hn(vox).
dx 3
(X + x0 3)1
1
From the asymptotic behaviors of Hn,
if
1,
Hn(vox)
=
Xn
if KnQp < 1 or xo > 1;
4LI-nq
(B3)
xo « 1;
X0if
LI,
D~=
Dnj =
< 1 or xo > 1;
, if KnQp > 1;
4
one finds
KnQp
1
ifKKnP
QP
,
(>4
(B4)
ifX0<1
Numerical calculations show that Dnj is a reasonably smooth function of
Qp and
Xto.
The simplest Padi form of Dnj satisfying Eq. (B4) is thus
D
=
(B5)
LI.
4Lt
4LI_1 + KnQpLt
This simple approximation is found in excellent agreement with the numerical results. For
a wide range of ,nQ, and xo, the errors are within 5%.
One can then obtain
cn
1--
U2n
~Sr.,vo
19
L e+
±
=1
n=1
4
_
L&Xt.
(B6)
In order to make the usage of Eq. (B6) straightforward, good Pad4 approximations of
fg
and
fCP are also desirable.
Using the asymptotic behaviors at e
{-
dA
f
4
1
1.6c3/2
<
1,
(B7)
1 - 1.46Ji/2
-A.dA
and at c -- 1
3
Ac
dA (
3f A
1.5(1
-
(B8)
i
Eqs. (21)-(22) yield the Pad4 approximations for
(Cd
c-
fg
and
(1-e 2 )-1/ 2 (1_c2 )
-1+1.60/f2-1.25C2
f1'
(9 9
(
1p (1-e2)-1/2(j_f)2
1+1.46E1/ 2 +0.2e
Here, (1 - C2)-1/2 comes from (h- 2 ) in a concentric circular equilibrium. The results
of
fg
and
ft',
calculated using both the Pad4 form Eq. (B9) and the exact numerical
integration of Eqs. (21)-(22), are given in Fig. (1). It shows the excellent accuracy of this
Pad6 approximation.
Furthermore, by using Eq. (B9), Eq. (B6) is evaluated and compared with the numerical evaluations (using the eigenfunctions A, for n = 1,2,3,4,5 obtained in Ref. 4).
For a wide range of Qp, Xo, and e, the differences are always within 10%, and the analytic approximate results are always larger than the numerical results (which is expected
owing to the fact that only eigenfunctions with n < 5 have been included in the numerical calculations). This shows that the Pad forms given in Eqs. (B6) and (B9) are quite
accurate.
20
References
1 K.
Tomabechi, Plas. Phys. Contr., Nucl. Fusion Res. 3, 214, IAEA, Vienna, 1989.
2 A.
Nocentini, M. Tesarotto, F. Engelmann, Nucl. Fusion 15, 359 (1975).
3 V.
Ya. Goroboroko, Ya. I. Kolesnichenko, V. A. Yavorskij, Nucl. Fusion 23, 399 (1983).
4 C.
T. Hsu, P. J. Catto, D. J. Sigmar, Phys. Fluids B2, 280 (1990).
S. P. Hirshman, D. J. Sigmar, Nucl. Fusion 21, 1079 (1881).
6C. T. Hsu, D. J. Sigmar, Plasma Phys. Contr. Fus. 32, 499 (1990).
7 J.
W. Connor, J. G. Cordey, Nucl. Fusion 14, 185 (1974).
8 L.
Spitzer, R. Hiirm, Phys. Rev. 89 977 (1953).
9 R.
D. Hazeltine, A. A. Ware, Plasma Phys. 20, 673 (1978).
' 0 F. L. Hinton, R. D. Hazeltine, Rev. Mod. Phys. 48, 239 (1976).
21
Zeff = 1.5
Zeff = 2.2
Zeff(r)
(flat)
(flat)
peaked
19.03
18.64
18.90
(MA)
5.34
5.05
5.23
I(a) (MA)
0.385
0.303
0.372
Itot (MA)
1(0)
Table 1. Induced bootstrap currents for three different
choices of Zeff. Input parameters are for the
steady state ITER machine. Ohmic seed current
is fixed at 13 MA.
22
pitch
angle
Fuel
scattering
dilution
switched of f
switched off
18.96
19.33
(o) (MA)
5.36
5.54
I(a) (MA)
0.294
0.471
Itot (MA)
1
Table 2.
Induced bootstrap currents (for Zeff = 1.5) produced after the effects of pitch angle scattering
and fuel dilution are individually switched
off.Input parameters are for the steady state
ITER machine. Ohmic seed current is fixed at 13
MA.
23
Figure Captions
Fig. 1:
Circulating particle fraction
fc
in the "pitch angle scattering" dominant and
"drag" dominant cases. Pad6 approximations are in excellent agreement with
the exact solution.
Fig. 2:
Effective fraction of trapped alphas, assuming four different birth energies Eo.
Fig. 1:
Poloidal magnetic field profile, with and without the effects of alpha bootstrap
current. Zeff = 1.5.
Fig. 4:
Alpha particle induced bootstrap current density as a function of r/a. Zff = 1.5.
Fig. 5:
Background bootstrap current density as a function of r/a. Zeff = 1.5.
Fig. 6:
Fraction of alpha bootstrap current density over background bootstrap current
density, for flat Zeff = 1.5 and centrally peaked Zqff (r) = 0.7 exp(-16r 2 /a 2 )+
1.5.
24
1.0
.
exact
.-
Pade Approximate
0.8
-S
0.6
fed
a
4.4
0.4
fCP
nfl'.
l-
0.0
0.2
0.6
0.4
r/R0
Figure 1
25
0.8
1.0
600
---- 2--
500-
E0 =
E0 =
-3-E 0 =
-- 4- Eo =
0.800 MeV
1.500 MeV
3.500 MeV
infinity
400 -
-
300 -
S/
-f
3000
200
0.0
0.2
0.6
0.4
r/a
Figure 2
26
0.8
1.0
1.4
I'
I
.
I-
I
bootstrapalpha
with
I
I
-
with alpha bootstrapno alpha bootstrap
I
I
1.2
1.0
-
0.8
------
0.6
0.4
-
-
0.2
i
0.0
0.2
0.4
0.6
radius r/a
Figure 3
27
0.8
1.0
30
25-
20-
15
15 ~10--
0
-5
0.0
'
'
I
0.2
0.4
0.6
Radius r/a
Figure 4
28
0.8
1.0
300
250
200-
150
100
50-
0
0.0
0.2
0.4
0.6
Radius r/a
Figure 5
29
0.8
1.0
150
100-
50
OM.
/
0-
Zert =
--
1.5
Zert = Zerr(r/a)
-50
0.0
0.2
0.4
0.6
radius r/a
Figure 6
30
0.8
1.0